Cracking RSA

RSA is a widely used public-key cryptosystem that allows two
parties (such as people or computers) to exchange secret
messages without revealing information to anyone else listening
on the conversation. Many websites use RSA when visited over a
secure https connection. In RSA, each
party has two different keys: a public key that is
published and a private key that is kept secret. To
encrypt a message intended for a specific recipient, the sender
will use the recipient’s public key to encrypt the message. The
recipient will use their private key to decrypt the
message.

An RSA public key $(n,
e)$ consists of two numbers $n$ and $e$. The number $n$ is a product of two distinct prime
numbers, $p$ and
$q$. In real applications,
$n$ would be hundreds of
decimal digits long for security.

Let $\varphi (n)$ be
Euler’s totient function, which in this case is equal
to $(p-1) (q-1)$. The
private key consists of $(n,
d)$, where $n$ is
the same as in the public key and $d$ is the solution to the
congruence

\[ de \equiv 1
\bmod \varphi (n) \]

Formally, a congruence

\[ a
\equiv b \bmod c \]

holds for three integers $a$, $b$, and $c$, if there exists an integer
$k$ such that $a - b = k c$.

The sender will encrypt a message $M$ (which, for simplicity, is assumed
to be an integer smaller than both $p$ and $q$) by computing $M^ e \bmod n$ and sending it to the
receiver. The recipient will calculate $(M^ e)^ d \equiv M^{e d} \equiv M^{k \varphi
(n) + 1} \equiv M^{\varphi (n) k} M \equiv M \bmod n$
since by Euler’s theorem $M^{\varphi (n)} \equiv 1 \bmod n$.
This will reconstruct the original message. Without the private
key, no practical way has been found for a potential attacker
to recover $M$ from the
knowledge of $M^ e \bmod
n$ and $(n,
e)$.

Your task is to crack RSA by finding the private key related
to a specific public key.

Input

The first line of input has the number of test cases
$T$, ($1 \le T \le 50$). Each test case has
one line that contains the two numbers $n$ and $e$. You may assume that $n$ is the product of two primes
$p, q$ such that
$2 \le p,q < 1000$.
Also, $e$ will be chosen
so that $d$ exists and is
unique, and $1 < d, e <
\varphi (n)$. Note that the product $d e$ may not fit into a 32-bit
integer (e.g. Java’s int type).